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Transcript
Chapter 7
Eigenvalues and Eigenvectors
7.1 Eigenvalues and eigenvectors
• Eigenvalue problem: If A
is an nn matrix, do there
exist nonzero vectors x in
Rn such that Ax is a scalar
multiple of x
7-1
7-2
• Note:
Ax  x  (I  A) x  0 (homogeneous system)
If (I  A) x  0 has nonzero solutions iff det(I  A)  0.
• Characteristic polynomial of AMnn:
det(I  A)  (I  A)  n  cn 1n 1    c1  c0
• Characteristic equation of A:
det(I  A)  0
7-3
7-4
• Notes:
(1)
If an eigenvalue 1 occurs as a multiple root (k times) for
the characteristic polynomial, then 1 has multiplicity k.
(2) The multiplicity of an eigenvalue is greater than or equal
to the dimension of its eigenspace.
7-5
7-6
• Eigenvalues and eigenvectors of linear transformations:
A number  is called an eigenvalue of a linear tra nsformatio n
T : V  V if there is a nonzero vector x such that T (x)  x.
The vector x is called an eigenvecto r of T correspond ing to  ,
and the setof all eigenvecto rs of  (with the zero vector) is
called the eigenspace of .
7-7
7.2 Diagonalization
• Diagonalization problem: For a square matrix A, does there
exist an invertible matrix P such that P-1AP is diagonal?
• Notes:
(1) If there exists an invertible matrix P such that B  P 1 AP ,
then two square matrices A and B are called similar.
(2) The eigenvalue problem is related closely to the
diagonalization problem.
7-8
7-9
7-10
7-11
7-12
7-13
7.3 Symmetric Matrices and Orthogonal
7-14
• Note: Theorem 7.7 is called the Real Spectral Theorem, and the
set of eigenvalues of A is called the spectrum of A.
7-15
7-16
7-17
• Note: A matrix A is orthogonally diagonalizable if there exists
an orthogonal matrix P such that P-1AP = D is diagonal.
7-18
7-19
7.4 Applications of Eigenvalues and
Eigenvectors
7-20
• If A is not diagonal:
-- Find P that diagonalizes A:
y  Pw
 y '  Pw '  Pw '  y '  Ay  APw
1
 w '  P APw
7-21
• Quadratic Forms
a' and c ' are eigenvalues of the matrix:
matrix of the quadratic form
7-22
7-23
7-24